Abstract We prove that the rational cohomology group $H^{11}(\overline {\mathcal {M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$ . show furthermore $H^k(\overline is pure Hodge–Tate for all even $k \leq 12$ deduce $\# \overline {M}}_{g,n}(\mathbb {F}_q)$ surprisingly well approximated by a polynomial in q In addition, we use {M}}_{1,11})$ its image under Gysin push-forward tautological map...